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Claim. Przemek Chojecki's post of 16 April 2026 in the discussion thread of Problem 326 links a short note with a counterexample that the post credits to GPT-5.4. The erdosproblems.com wording then asked Erdős and Graham's "another way of stating the problem": does every basis of order $2$ have a subset which is also a basis and for which $\lim a_k/k^2$ does not exist? The note answers no. Its basis is the set of natural numbers whose ternary digits are all $0$ or $1$, a basis of order $2$ since the sum of two such numbers can have any ternary digit, and every sub-basis $\{b_k\}$ of it has $b_k/k^2\to0$, so the limit exists for every sub-basis.

Submission note. Posted to the erdosproblems.com forum by Przemek Chojecki on 16 April 2026:

Looks like this is false. A counterexample given by GPT-5.4 is

$$> A = \Bigl\{ \sum_{j\ge 0} \varepsilon_j 3^j : \varepsilon_j\in\{0,1\}, > \ \text{for all but finitely many j} \Bigr\} >$$

Thus $A$ is the set of natural numbers whose ternary digits are all 0 or 1 (alternatively exclude 0 depending on the convention). Here's a short note with a proof that this is indeed a counterexample

Why it is rejected. It answers a variant, not the Statement. The note answers erdosproblems.com's earlier wording, in which a limit $0$ of $a_k/k^2$ counts as a limit. The erdosproblems.com curator replied the same day that the limit was meant to be a constant $c\ne0$, since otherwise there are easy counterexamples such as this one, and rewrote the problem as Erdős and Graham's minimal-basis question, the Statement of Problem 326: is there a minimal basis of order $2$ with $a_k/k^2\to c$ for some $c\ne0$? A basis whose sub-bases all have $b_k/k^2\to0$ has no such sub-basis and says nothing about minimal bases with a nonzero limit, so its mathematics answers the variant and does not settle this problem; the value disproved is its answer to that variant.

Standing. Rejected, as answering a variant. The note is unrefereed, the thread records no review of it, and this corpus has not checked it.

Depends on. No page of this wiki.